\documentclass{article}
\usepackage{fullpage}


% If your system does not have the AMS fonts version 2.0 installed, then
% remove the useAMS option.
%
% useAMS allows you to obtain upright Greek characters.
% e.g. \umu, \upi etc.  See the section on "Upright Greek characters" in
% this guide for further information.
%
% If you are using AMS 2.0 fonts, bold math letters/symbols are available
% at a larger range of sizes for NFSS release 1 and 2 (using \boldmath or
% preferably \bmath).
%
% The usenatbib command allows the use of Patrick Daly's natbib.sty for
% cross-referencing.
%
% If you wish to typeset the paper in Times font (if you do not have the
% PostScript Type 1 Computer Modern fonts you will need to do this to get
% smoother fonts in a PDF file) then uncomment the next line
% \usepackage{Times}

%%%%% AUTHORS - PLACE YOUR OWN MACROS HERE %%%%%


%% cosmology

\newcommand{\ns}{n_{\rm s} }
\newcommand{\guv}{g_{\mu \nu} }

%% ions / absorbers


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{Cosmology Notes}
\author{Gabriel Altay}
\begin{document}
\maketitle


\section{Introduction}

These are my notes on cosmology.  For general relativity, I will use the sign conventions of Misner, Thorne, and Wheeler.  In particular, the Minkowski metric $\eta_{\mu \nu}$ has signature $(-,+,+,+)$.  I will follow the common convention of using greek letters for space-time indices $\alpha, \beta, \nu = 0,1,2,3$ and latin letters for spatial indices $i,j = 1,2,3$. 



\section{General Relativity - Basics}

The fundamental object describing the curvature of spacetime in general relativity is the metric $g_{\mu \nu}$.  It defines the invariant interval $ds$ between two spacetime events with coordinate separation $dx^{\alpha}$
\begin{equation}
ds^2 = g_{\mu \nu} dx^\mu dx^\nu
\end{equation}


\section{Perfect Fluid Universes}

To begin we consider models in which the mass-energy content of the Universe 
is perfectly smooth.  This is an idealization of the cosmological principle which is the working assumption that the universe if both isotropic and homogeneous on large scales.  Under this assumption the invariant interval  must take the form, 
\begin{equation}
ds^2 = -c^2 dt^2 + a(t)^2 d \Sigma^2
\end{equation} 


\subsection{Cartesian Coordinates}

In cartesian $(x^0,x^1,x^2,x^3) = (t,x,y,z)$ coordinates $d \Sigma^2 = dx^2 + dy^2 + dz^2$.  The components of the metric are 
\begin{equation}
g_{00} = -c^2, \quad g_{11} = a^2, \quad g_{22} = a^2, \quad g_{33} = a^2
\end{equation}
\begin{equation}
g^{00} = -c^{-2}, \quad g^{11} = a^{-2}, \quad g^{22} = a^{-2}, \quad g^{33} = a^{-2}
\end{equation}
\begin{equation}
\frac{da^2}{dt} = \dot{a} a + a \dot{a} = 2 \dot{a} a
\end{equation} 

\begin{equation}
\Gamma_{\alpha \beta}^{\mu} = \frac{1}{2} g^{\mu \nu}
\left[ 
\frac{\partial g_{\alpha \nu}}{\partial x^\beta} +
\frac{\partial g_{\beta \nu}}{\partial x^\alpha} -
\frac{\partial g_{\alpha \beta}}{\partial x^\nu} 
\right]
\end{equation}

\begin{equation}
\Gamma_{0 0}^{0} = \frac{1}{2} g^{0 \nu}
\left[ 
\frac{\partial g_{0 \nu}}{\partial x^0} +
\frac{\partial g_{0 \nu}}{\partial x^0} -
\frac{\partial g_{0 0}}{\partial x^\nu} 
\right] = 
\frac{1}{2} g^{0 0}
\left[ 
\frac{\partial g_{0 0}}{\partial x^0} +
\frac{\partial g_{0 0}}{\partial x^0} -
\frac{\partial g_{0 0}}{\partial x^0} 
\right] = 0
\end{equation}

\begin{equation}
\Gamma_{0 j}^{0} = \Gamma_{j 0}^{0} = \frac{1}{2} g^{0 \nu}
\left[ 
\frac{\partial g_{j \nu}}{\partial x^0} +
\frac{\partial g_{0 \nu}}{\partial x^j} -
\frac{\partial g_{j 0}}{\partial x^\nu} 
\right] = 
\frac{1}{2} g^{0 0}
\left[ 
\frac{\partial g_{j 0}}{\partial x^0} +
\frac{\partial g_{0 0}}{\partial x^j} -
\frac{\partial g_{j 0}}{\partial x^0} 
\right] = 0
\end{equation}


\begin{equation}
\Gamma_{i j}^{0} = \frac{1}{2} g^{0 \nu}
\left[ 
\frac{\partial g_{i \nu}}{\partial x^j} +
\frac{\partial g_{j \nu}}{\partial x^i} -
\frac{\partial g_{i j}}{\partial x^\nu} 
\right] = 
\frac{1}{2} g^{0 0}
\left[ 
\frac{\partial g_{i 0}}{\partial x^j} +
\frac{\partial g_{j 0}}{\partial x^i} -
\frac{\partial g_{i j}}{\partial x^0} 
\right] = 
\frac{1}{-2c^2}
\left[ 
\frac{\partial g_{i j}}{\partial t} 
\right] = 
-\frac{\dot{a} a}{c^2} \delta_{ij}
\end{equation}

\begin{equation}
\Gamma_{0 0}^{i} = \frac{1}{2} g^{i \nu}
\left[ 
\frac{\partial g_{0 \nu}}{\partial x^0} +
\frac{\partial g_{0 \nu}}{\partial x^0} -
\frac{\partial g_{0 0}}{\partial x^\nu} 
\right] = 
\frac{1}{2} g^{i i}
\left[ 
\frac{\partial g_{0 i}}{\partial x^0} +
\frac{\partial g_{0 i}}{\partial x^0} -
\frac{\partial g_{0 0}}{\partial x^i} 
\right] = 0
\end{equation}


\begin{equation}
\Gamma_{0 j}^{i} = \Gamma_{j 0}^{i} = 
\frac{1}{2} g^{i \nu}
\left[ 
\frac{\partial g_{j \nu}}{\partial x^0} +
\frac{\partial g_{0 \nu}}{\partial x^j} -
\frac{\partial g_{j 0}}{\partial x^\nu} 
\right] = 
\frac{1}{2} g^{i i}
\left[ 
\frac{\partial g_{j i}}{\partial x^0} +
\frac{\partial g_{0 i}}{\partial x^j} -
\frac{\partial g_{j 0}}{\partial x^i} 
\right] = 
\frac{1}{2a^2}
\left[ 
\frac{\partial g_{j i}}{\partial t} 
\right] = 
\frac{\dot{a}}{a} \delta_{ij}
\end{equation}


\begin{equation}
\Gamma_{i j}^{i} = \Gamma_{j i}^{i} = 
\frac{1}{2} g^{i \nu}
\left[ 
\frac{\partial g_{j \nu}}{\partial x^i} +
\frac{\partial g_{i \nu}}{\partial x^j} -
\frac{\partial g_{j i}}{\partial x^\nu} 
\right] = 
\frac{1}{2} g^{i i}
\left[ 
\frac{\partial g_{j i}}{\partial x^i} +
\frac{\partial g_{i i}}{\partial x^j} -
\frac{\partial g_{j i}}{\partial x^i} 
\right] =  0
\end{equation}



\begin{equation}
R_{\mu \nu} = 
\Gamma_{\mu \nu,\alpha}^{\alpha} - 
\Gamma_{\mu \alpha,\nu}^{\alpha} + 
\Gamma_{\mu \nu}^{\alpha} \Gamma_{\alpha \beta}^{\alpha} - 
\Gamma_{\mu \beta}^{\alpha} \Gamma_{\alpha \nu}^{\beta} 
\end{equation}




\subsection{Spherical Coordinates}
 or $d \Sigma^2 = dr^2 + r^2 (d \theta^2 + \sin^2\theta d \phi^2) $ in spherical coordinates.  In terms of metric elements we have, 
\begin{equation}
dx^0 = dt, \quad dx^1 = dr, \quad 
dx^2 = d\theta, \quad dx^3 = d\phi
\end{equation} 
\begin{equation}
g_{00} = -c^2, \quad g_{11} = a^2, \quad 
g_{22} = a^2 r^2, \quad g_{33} = a^2 r^2 \sin^2 \theta
\end{equation} 
\begin{equation}
g^{00} = \frac{1}{-c^{2}}, \quad g^{11} = \frac{1}{a^{2}}, \quad 
g^{22} = \frac{1}{a^{2} r^{2}}, \quad g^{33} = \frac{1}{a^{2} r^{2} \sin^{2} \theta}
\end{equation} 


\begin{equation}
\Gamma_{\alpha \beta}^{\mu} = \frac{1}{2} g^{\mu \nu}
\left[ 
\frac{\partial g_{\alpha \nu}}{\partial x^\beta} +
\frac{\partial g_{\beta \nu}}{\partial x^\alpha} -
\frac{\partial g_{\alpha \beta}}{\partial x^\nu} 
\right]
\end{equation}

\begin{equation}
\Gamma_{i j}^{0} = \frac{1}{2} g^{0 \nu}
\left[ 
\frac{\partial g_{i \nu}}{\partial x^j} +
\frac{\partial g_{j \nu}}{\partial x^i} -
\frac{\partial g_{i j}}{\partial x^\nu} 
\right] = 
\frac{1}{2} g^{0 0}
\left[ 
\frac{\partial g_{i 0}}{\partial x^j} +
\frac{\partial g_{j 0}}{\partial x^i} -
\frac{\partial g_{i j}}{\partial x^0} 
\right] = 
\frac{1}{-2c^2}
\left[ 
\frac{\partial g_{i j}}{\partial t} 
\right]
\end{equation}


\begin{equation}
\Gamma_{3 3}^{2} = \frac{1}{2} g^{2 \nu}
\left[ 
\frac{\partial g_{3 \nu}}{\partial x^3} +
\frac{\partial g_{3 \nu}}{\partial x^3} -
\frac{\partial g_{3 3}}{\partial x^\nu} 
\right]
\end{equation}


\section{Linear Theory}

Modern cosmology is based on solutions to Einstein's field equations,
\begin{equation}
G_{\mu \nu} = 
R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R + g_{\mu \nu} \Lambda  = 
\frac{8 \pi G}{c^4} T_{\mu \nu}
\end{equation} 
where $G_{\mu \nu}$ is the Einstein tensor, $R_{\mu \nu}$ is the Ricci curvature tensor, $\guv$ is the metric tensor, $R$ is the Ricci scalar, $\Lambda$ is the cosmological constant, $G$ is Newton's gravitational constant, $c$ is the speed of light, and $T_{\mu \nu}$ is the stress-energy tensor.  

The Christoffel symbols are defined as, 
\begin{equation}
\Gamma_{\alpha \beta}^{\mu} = \frac{1}{2} g^{\mu \nu}
\left[ 
\frac{\partial g_{\alpha \nu}}{\partial x^\beta} +
\frac{\partial g_{\beta \nu}}{\partial x^\alpha} -
\frac{\partial g_{\alpha \beta}}{\partial x^\nu} 
\right]
\end{equation}
We can examine some special cases.  In the case that $\mu=0$,

\begin{equation}
\Gamma_{\alpha \beta}^{0} = \frac{1}{2} g^{0 \nu}
\left[ 
\frac{\partial g_{\alpha \nu}}{\partial x^\beta} +
\frac{\partial g_{\beta \nu}}{\partial x^\alpha} -
\frac{\partial g_{\alpha \beta}}{\partial x^\nu} 
\right]
\end{equation}

The first solution we will discuss is that due to Friedmann, Lema\^{i}tre, Robertson, and Walker.  In this model the constituents of the Universe are perfect fluids meaning that $T_{\mu \nu} = {\rm diag} (\rho, P, P, P) $    

\begin{equation}
ds^2 = -dt^2 + a^2(t) \eta_{ik} dx^i dx^k
\end{equation}

In cosmological perturbation theory one considers a perturbed space-time metric of the form, 

\begin{equation}
  \guv = \bar{g}_{\mu \nu} + \delta \guv = a^2( \eta_{\mu \nu} + h_{\mu \nu} ) 
\end{equation}
where $\bar{g}_{\mu \nu}$ is the smooth background metric and $\delta \guv$ represents small perturbations.  In the flat FLRW cosmological model the background metric is proportional to the Minkowski metric $\eta_{\mu \nu}$
and the perturbations are quantified using $h_{\mu \nu}$.

The primoridial power spectrum, leaving out normalization for now, 
\begin{equation}
  P_{\infty} = P(k,\infty) = k^{\ns}
\end{equation}


\begin{equation}
  P(k,z) = k^{n_s} T(k) D_1(z)
\end{equation}

\begin{equation}
  \sigma^2(R,z) = \int_0^\infty 
  k^2 \frac{P(k)}{2 \pi^2}
\end{equation}


\bibliographystyle{alpha} % or "unsrt", "alpha", "abbrv", etc.
\bibliography{biblio}	  % use data in file "astrobibl.bib"


\end{document}
